From Surf Wiki (app.surf) — the open knowledge base
Biggs–Smith graph
Cubic distance-regular graph with 102 nodes and 153 edges
Cubic distance-regular graph with 102 nodes and 153 edges
| Field | Value | |
|---|---|---|
| name | Biggs–Smith graph | |
| image | [[Image:Biggs-Smith graph.svg | 250px]] |
| image_caption | The Biggs–Smith graph | |
| vertices | 102 | |
| edges | 153 | |
| automorphisms | 2448 (PSL(2,17)) | |
| girth | 9 | |
| radius | 7 | |
| diameter | 7 | |
| chromatic_number | 3 | |
| chromatic_index | 3 | |
| properties | Symmetric | |
| Distance-regular | ||
| Cubic | ||
| Hamiltonian |
Distance-regular Cubic Hamiltonian
In the mathematical field of graph theory, the Biggs–Smith graph is a 3-regular graph with 102 vertices and 153 edges.
It has chromatic number 3, chromatic index 3, radius 7, diameter 7 and girth 9. It is also a 3-vertex-connected graph and a 3-edge-connected graph.
All the cubic distance-regular graphs are known. The Biggs–Smith graph is one of the 13 such graphs.
Algebraic properties
The automorphism group of the Biggs–Smith graph is a group of order 2448 isomorphic to the projective special linear group PSL(2,17). It acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore, the Biggs–Smith graph is a symmetric graph. It has automorphisms that take any vertex to any other vertex and any edge to any other edge. According to the Foster census, the Biggs–Smith graph, referenced as F102A, is the only cubic symmetric graph on 102 vertices.
The Biggs–Smith graph is also uniquely determined by its graph spectrum, the set of graph eigenvalues of its adjacency matrix.
The characteristic polynomial of the Biggs–Smith graph is : (x-3) (x-2)^{18} x^{17} (x^2-x-4)^9 (x^3+3 x^2-3)^{16}.
Gallery
Image:Biggs-Smith graph 3COL.svg|The chromatic number of the Biggs–Smith graph is 3. Image:Biggs-Smith graph 3color edge.svg|The chromatic index of the Biggs–Smith graph is 3. File:Biggs-Smith graph unit distance.svg|The Biggs–Smith graph is a unit-distance graph. Image:Biggs-Smith graph - circle-based drawing.jpg|The Biggs–Smith graph is an order-17 graph expansion of the H graph.
References
- On trivalent graphs, NL Biggs, DH Smith - Bulletin of the London Mathematical Society, 3 (1971) 155–158.
References
- "Biggs–Smith Graph".
- [[A. E. Brouwer. Brouwer, A. E.]]; Cohen, A. M.; and Neumaier, A. Distance-Regular Graphs. New York: Springer-Verlag, 1989.
- "G-17 Biggs-Smith graph".
- [[Marston Conder. Conder, M.]] and Dobcsányi, P. "Trivalent Symmetric Graphs Up to 768 Vertices." J. Combin. Math. Combin. Comput. 40, 41–63, 2002.
- E. R. van Dam and W. H. Haemers, Spectral Characterizations of Some Distance-Regular Graphs. J. Algebraic Combin. 15, pages 189–202, 2003
This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.
Ask Mako anything about Biggs–Smith graph — get instant answers, deeper analysis, and related topics.
Research with MakoFree with your Surf account
Create a free account to save articles, ask Mako questions, and organize your research.
Sign up freeThis content may have been generated or modified by AI. CloudSurf Software LLC is not responsible for the accuracy, completeness, or reliability of AI-generated content. Always verify important information from primary sources.
Report